Problem: Simplify the following expression: $ a = \dfrac{5x - 9}{9x + 4} + \dfrac{-5}{7} $
Solution: In order to add expressions, they must have a common denominator. Multiply the first expression by $\dfrac{7}{7}$ $ \dfrac{5x - 9}{9x + 4} \times \dfrac{7}{7} = \dfrac{35x - 63}{63x + 28} $ Multiply the second expression by $\dfrac{9x + 4}{9x + 4}$ $ \dfrac{-5}{7} \times \dfrac{9x + 4}{9x + 4} = \dfrac{-45x - 20}{63x + 28} $ Therefore $ a = \dfrac{35x - 63}{63x + 28} + \dfrac{-45x - 20}{63x + 28} $ Now the expressions have the same denominator we can simply add the numerators: $a = \dfrac{35x - 63 - 45x - 20}{63x + 28} $ $a = \dfrac{-10x - 83}{63x + 28}$